Abstract

It is well known that bichromatic excitation on one common transition can tune the emission or absorption spectra of atoms due to the modulation frequency dependent non-linearities. However little attention has been focused on the quantum dynamics of fields under bichromatic excitation. Here we present dissipative effects on noise correlations of fields in bichromatic interactions with atoms in cavities. We first consider an ensemble of two-level atoms that interacts with the two cavity fields of different frequencies and considerable amplitudes. By transferring the atom-field nonlinearities to the dressed atoms we separate out the dissipative interactions of Bogoliubov modes with the dressed atoms. The Bogoliubov mode dissipation establishes stable two-photon processes of two involved fields and therefore leads to two-mode squeezing. As a generalization, we then consider an ensemble of three-level Λ atoms for cascade bichromatic interactions. We extract the Bogoliubov-like four-mode interactions, which establish a quadrilateral of the two-photon processes of four involved fields and thus result in four-mode squeezing.

© 2016 Optical Society of America

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References

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  34. Z. Ficek, J. Seke, A. V. Soldatov, G. Adam, and N. N. Bogoubov, “Multilevel coherence effects in a two-level atom driven by a trichromatic field,” Opt. Commun. 217, 299–309 (2003).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  52. G. S. Agarwal, W. Lange, and H. Walther, “Intense-field renormalization of cavity-induced spontaneous emission,” Phys. Rev. A 48, 4555–4568 (1993).
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    [Crossref]

2015 (1)

X. M. Hu, “Entanglement generation by dissipation in or beyond dark resonances,” Phys. Rev. A 92, 022329 (2015).
[Crossref]

2013 (1)

E. G. Dalla Torre, J. Otterbach, E. Demler, V. Vuletic, and M. D. Lukin, “Dissipative preparation of spin squeezed atomic ensembles in a steady state,” Phys. Rev. Lett. 110, 120402 (2013).
[Crossref] [PubMed]

2012 (1)

X. Liang, X. M. Hu, and C. He, “Creating multimode squeezed states and Greenberger-Horne-Zeilinger entangled states using atomic coherent effects,” Phys. Rev. A 85, 032329 (2012).
[Crossref]

2011 (1)

H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, J. M. Petersen, J. I. Cirac, and E. S. Polzik, “Entanglement generated by dissipation and steady state entanglement of two macroscopic objects,” Phys. Rev. Lett. 107, 080503 (2011).
[Crossref] [PubMed]

2010 (1)

X. Zhang and X. M. Hu, “Entanglement between collective fields via atomic coherence effects,” Phys. Rev. A 81, 013811 (2010).
[Crossref]

2009 (1)

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009).
[Crossref]

2008 (2)

P. B. Li, “Generation of two-mode field squeezing through selective dynamics in cavity QED,” Phys. Rev. A 77, 015809 (2008).
[Crossref]

G. L. Cheng, X. M. Hu, W. X. Zhong, and Q. Li, “Two-channel interaction of squeeze-transformed modes with dressed atoms: entanglement enhancement in four-wave mixing in three-level systems,” Phys. Rev. A 78, 033811 (2008).
[Crossref]

2007 (1)

S. Pielawa, G. Morigi, D. Vitali, and L. Davidovich, “Generation of Einstein-Podolsky-Rosen-entangled radiation through an atomic reservoir,” Phys. Rev. Lett. 98, 240401 (2007).
[Crossref] [PubMed]

2006 (5)

A. Dantan, J. Cviklinski, E. Giacobino, and M. Pinard, “Spin squeezing and light entanglement in coherent population trapping,” Phys. Rev. Lett. 97, 023605 (2006).
[Crossref] [PubMed]

A. Sinatra, “Quantum correlations of two optical fields close to electromagnetically induced transparency,” Phys. Rev. Lett. 97, 253601 (2006).
[Crossref]

R. Guzmán, J. C. Retamal, E. Solano, and N. Zagury, “Field squeeze operators in optical cavities with atomic ensembles,” Phys. Rev. Lett. 96, 010502 (2006).
[Crossref] [PubMed]

A. S. Parkins, E. Solano, and J. I. Cirac, “Unconditional two-mode squeezing of separated atomic ensembles,” Phys. Rev. Lett. 96, 053602 (2006).
[Crossref] [PubMed]

X. M. Hu, Q. Xu, J. Y. Li, X. X. Li, W. X. Shi, and X. Zhang, “Bichromatic and trichromatic manipulation of spontaneous emission in a three-level system,” Opt. Commun. 260, 196–202 (2006).
[Crossref]

2005 (5)

X. M. Hu, J. H. Zou, X. Li, D. Du, and G. L. Cheng, “Amplitude and phase control of trichromatic electromagnetically induced transparency,” J. Phys. B: At. Mol. Opt. Phys. 38, 683–692 (2005).
[Crossref]

X. M. Hu, G. L. Cheng, J. H. Zou, X. Li, and D. Du, “Double switching from normal to anomalous dispersion via trichromatic phase manipulation of electromagnetically induced transparency,” Phys. Rev. A 72, 023803 (2005).
[Crossref]

J. H. Zou, X. M. Hu, G. L. Cheng, X. Li, and D. Du, “Inhibition of two-photon absorption in a three-level system with a pair of bichromatic fields,” Phys. Rev. A 72, 055802 (2005).
[Crossref]

V. A. Sautenkov, Y. V. Rostovtsev, and M. O. Scully, “Switching between photon-photon correlations and Raman anticorrelations in a coherently prepared Rb vapor,” Phys. Rev. A 72, 065801 (2005).
[Crossref]

S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
[Crossref]

2004 (2)

P. Barberis-Blostein and N. Zagury, “Field correlations in electromagnetically induced transparency,” Phys. Rev. A 70, 053827 (2004).
[Crossref]

Y. Wu, M. G. Payne, E. W. Hagley, and L. Deng, “Preparation of multiparty entangled states using pairwise perfectly efficient single-probe photon four-wave mixing,” Phys. Rev. A 69, 063803 (2004).
[Crossref]

2003 (2)

Z. Ficek, J. Seke, A. V. Soldatov, G. Adam, and N. N. Bogoubov, “Multilevel coherence effects in a two-level atom driven by a trichromatic field,” Opt. Commun. 217, 299–309 (2003).
[Crossref]

J. Wang, Y. Zhu, K. J. Jiang, and M. S. Zhan, “Bichromatic electromagnetically induced transparency in cold rubidium atoms,” Phys. Rev. A 68, 063810 (2003).
[Crossref]

2002 (1)

D. L. Aronstein, R. S. Bennink, R. W. Boyd, and C. R. Stroud, “Comment on Resonance-fluorescence and absorption spectra of a two-level atom driven by a strong bichromatic field,” Phys. Rev. A 65, 067401 (2002).
[Crossref]

2001 (2)

Z. Ficek, J. Seke, A. V. Soldatov, and G. Adam, “Fluorescence spectrum of a two-level atom driven by a multiple modulated field,” Phys. Rev. A 64, 013813 (2001).
[Crossref]

J. R. Bochinski, C. C. Yu, T. Loftus, and T. W. Mossberg, “Vacuum-mediated multiphoton transitions,” Phys. Rev. A 63, 051402(R) (2001).
[Crossref]

2000 (2)

Z. Ficek, J. Seke, A. V. Soldatov, and G. Adam, “Phase control of subharmonic resonances,” Opt. Commun. 182, 143–150 (2000).
[Crossref]

L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[Crossref] [PubMed]

1999 (4)

Z. Ficek and T. Rudolph, “Quantum interference in a driven two-level atom,” Phys. Rev. A 60, R4245–R4248 (1999).
[Crossref]

A. D. Greentree, C. Wei, and N. B. Manson, “Polychromatic excitation of a two-level system,” Phys. Rev. A 59, 4083–4086 (1999).
[Crossref]

T. H. Yoon, M. S. Chung, and H. W. Lee, “Absorption spectra of two-level atoms interacting with a strong polychromatic pump field and an arbitrarily intense probe field,” Phys. Rev. A 60, 2547–2553 (1999).
[Crossref]

T. H. Yoon, S. A. Pulkin, J. R. Park, M. S. Chung, and H. W. Lee, “Theoretical analysis of resonances in the polarization spectrum of a two-level atom driven by a polychromatic field,” Phys. Rev. A 60, 605–613 (1999).
[Crossref]

1998 (3)

G. Yu. Kryuchkyan, M. Jakob, and A. S. Sargsian, “Resonance fluorescence in a bichromatic field as a source of nonclassical light,” Phys. Rev. A 57, 2091–2095 (1998).
[Crossref]

M. Jakob and G. Yu. Kryuchkyan, “Squeezing in the resonance fluorescence of a bichromatically driven two-level atom,” Phys. Rev. A 58, 767–770 (1998).
[Crossref]

A. Sinatra, J. F. Roch, K. Vigneron, Ph. Grelu, J. Ph. Poizat, K. Wang, and P. Grangier, “Quantum-nondemolition measurements using cold trapped atoms: comparison between theory and experiment,” Phys. Rev. A 57, 2980–2995 (1998).
[Crossref]

1997 (2)

C. C. Yu, J. R. Bochinski, T. M. V. Kordich, T. W. Mossberg, and Z. Ficek, “Driving the driven atom: spectral signatures,” Phys. Rev. A 56, R4381–R4384 (1997).
[Crossref]

P. B. Sellin, C. C. Yu, J. R. Bochinski, and T. W. Mossberg, “Intrinsically irreversible multiphoton laser gain mechanisms,” Phys. Rev. Lett. 78, 1432–1435 (1997).
[Crossref]

1996 (2)

M. F. Van Leeuwen, S. Papademetriou, and C. R. Stroud, “Autler-Townes effect for an atom in a 100% amplitude-modulated laser field. I. a dressed-atom approach,” Phys. Rev. A 53, 990–996 (1996).
[Crossref] [PubMed]

S. Papademetriou, M. F. Van Leeuwen, and C. R. Stroud, “Autler-Townes effect for an atom in a 100% amplitude-modulated laser field. II. experimental results,” Phys. Rev. A 53, 997–1003 (1996).
[Crossref] [PubMed]

1995 (1)

Yu. A. Zinin and N. V. Sushilov, “Absorption and dispersion spectra of a polychromatic field in a two-level medium driven by a strong polychromatic pumping field,” Phys. Rev. A 51, 3916–3921 (1995).
[Crossref] [PubMed]

1994 (1)

W. Lange, H. Walther, and G. S. Agarwal, “Decay of bichromatically driven atoms in a cavity,” Phys. Rev. A 50, R3593–R3596 (1994).
[Crossref]

1993 (2)

G. S. Agarwal, W. Lange, and H. Walther, “Intense-field renormalization of cavity-induced spontaneous emission,” Phys. Rev. A 48, 4555–4568 (1993).
[Crossref] [PubMed]

Z. Ficek and H. S. Freedhoff, “Resonance-Auorescence and absorption spectra of a two-level atom driven by a strong bichromatic field,” Phys. Rev. A 48, 3092–3104 (1993).
[Crossref] [PubMed]

1991 (1)

1990 (1)

Y. Zhu, Q. Wu, A. Lezama, D. J. Gauthier, and T. W. Mossberg, “Resonance fluorescence of tvvo-level atoms under strong bichromatic excitation,” Phys. Rev. A 41, R6574–R6576 (1990).
[Crossref]

1988 (1)

M. O. Scully, K. Wódkiewicz, M. S. Zubairy, J. Bergou, N. Lu, and J. Myer ter Vehn, “Two-Photon Correlated-Spontaneous-Emission Laser: Quantum Noise Quenching and Squeezing,” Phys. Rev. Lett. 60, 1832–1835 (1988).
[Crossref] [PubMed]

1985 (1)

M. D. Reid, D. F. Walls, and B. J. Dalton, “Squeezing of quantum fluctuations via atomic coherence effects,” Phys. Rev. Lett. 55, 1288–1290 (1985).
[Crossref] [PubMed]

1981 (1)

P. D. Drummond and D. F. Walls, “Quantum theory of optical bistability. II. atomic fluorescence in a high-Q cavity,” Phys. Rev. A 23, 2563–2579 (1981).
[Crossref]

1980 (3)

P. D. Drummond and C. W. Gardiner, “Generalised P-representations in quantum optics,” J. Phys. A 13, 2353–2368 (1980).
[Crossref]

C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sandberg, and Zimmermann, “On the rrieasureri-ient of a weak classical force coupled to a quantum-mechanical oscillator. I. Issues of principle,” Rev. Mod. Phys. 52, 341–392 (1980).
[Crossref]

B. Blind, P. R. Fontana, and P. Thomann, “Resonance fluorescence spectrum of intense amplitude modulated laser light,” J. Phys. B: At. Mol. Opt. Phys. 13, 2717–2727 (1980).
[Crossref]

1935 (1)

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[Crossref]

Adam, G.

Z. Ficek, J. Seke, A. V. Soldatov, G. Adam, and N. N. Bogoubov, “Multilevel coherence effects in a two-level atom driven by a trichromatic field,” Opt. Commun. 217, 299–309 (2003).
[Crossref]

Z. Ficek, J. Seke, A. V. Soldatov, and G. Adam, “Fluorescence spectrum of a two-level atom driven by a multiple modulated field,” Phys. Rev. A 64, 013813 (2001).
[Crossref]

Z. Ficek, J. Seke, A. V. Soldatov, and G. Adam, “Phase control of subharmonic resonances,” Opt. Commun. 182, 143–150 (2000).
[Crossref]

Agarwal, G. S.

W. Lange, H. Walther, and G. S. Agarwal, “Decay of bichromatically driven atoms in a cavity,” Phys. Rev. A 50, R3593–R3596 (1994).
[Crossref]

G. S. Agarwal, W. Lange, and H. Walther, “Intense-field renormalization of cavity-induced spontaneous emission,” Phys. Rev. A 48, 4555–4568 (1993).
[Crossref] [PubMed]

G. S. Agarwal, Y. Zhu, D. J. Gauthier, and T. W. Mossberg, “Spectrum of radiation from two-level atoms under intense bichromatic excitation,” J. Opt. Soc. Am. B 8, 1163–1167 (1991).
[Crossref]

Aronstein, D. L.

D. L. Aronstein, R. S. Bennink, R. W. Boyd, and C. R. Stroud, “Comment on Resonance-fluorescence and absorption spectra of a two-level atom driven by a strong bichromatic field,” Phys. Rev. A 65, 067401 (2002).
[Crossref]

Barberis-Blostein, P.

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S. Pielawa, G. Morigi, D. Vitali, and L. Davidovich, “Generation of Einstein-Podolsky-Rosen-entangled radiation through an atomic reservoir,” Phys. Rev. Lett. 98, 240401 (2007).
[Crossref] [PubMed]

Pinard, M.

A. Dantan, J. Cviklinski, E. Giacobino, and M. Pinard, “Spin squeezing and light entanglement in coherent population trapping,” Phys. Rev. Lett. 97, 023605 (2006).
[Crossref] [PubMed]

Podolsky, B.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[Crossref]

Poizat, J. Ph.

A. Sinatra, J. F. Roch, K. Vigneron, Ph. Grelu, J. Ph. Poizat, K. Wang, and P. Grangier, “Quantum-nondemolition measurements using cold trapped atoms: comparison between theory and experiment,” Phys. Rev. A 57, 2980–2995 (1998).
[Crossref]

Polzik, E. S.

H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, J. M. Petersen, J. I. Cirac, and E. S. Polzik, “Entanglement generated by dissipation and steady state entanglement of two macroscopic objects,” Phys. Rev. Lett. 107, 080503 (2011).
[Crossref] [PubMed]

Pulkin, S. A.

T. H. Yoon, S. A. Pulkin, J. R. Park, M. S. Chung, and H. W. Lee, “Theoretical analysis of resonances in the polarization spectrum of a two-level atom driven by a polychromatic field,” Phys. Rev. A 60, 605–613 (1999).
[Crossref]

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M. D. Reid, D. F. Walls, and B. J. Dalton, “Squeezing of quantum fluctuations via atomic coherence effects,” Phys. Rev. Lett. 55, 1288–1290 (1985).
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Retamal, J. C.

R. Guzmán, J. C. Retamal, E. Solano, and N. Zagury, “Field squeeze operators in optical cavities with atomic ensembles,” Phys. Rev. Lett. 96, 010502 (2006).
[Crossref] [PubMed]

Roch, J. F.

A. Sinatra, J. F. Roch, K. Vigneron, Ph. Grelu, J. Ph. Poizat, K. Wang, and P. Grangier, “Quantum-nondemolition measurements using cold trapped atoms: comparison between theory and experiment,” Phys. Rev. A 57, 2980–2995 (1998).
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A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[Crossref]

Rostovtsev, Y. V.

V. A. Sautenkov, Y. V. Rostovtsev, and M. O. Scully, “Switching between photon-photon correlations and Raman anticorrelations in a coherently prepared Rb vapor,” Phys. Rev. A 72, 065801 (2005).
[Crossref]

Rudolph, T.

Z. Ficek and T. Rudolph, “Quantum interference in a driven two-level atom,” Phys. Rev. A 60, R4245–R4248 (1999).
[Crossref]

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C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sandberg, and Zimmermann, “On the rrieasureri-ient of a weak classical force coupled to a quantum-mechanical oscillator. I. Issues of principle,” Rev. Mod. Phys. 52, 341–392 (1980).
[Crossref]

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G. Yu. Kryuchkyan, M. Jakob, and A. S. Sargsian, “Resonance fluorescence in a bichromatic field as a source of nonclassical light,” Phys. Rev. A 57, 2091–2095 (1998).
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V. A. Sautenkov, Y. V. Rostovtsev, and M. O. Scully, “Switching between photon-photon correlations and Raman anticorrelations in a coherently prepared Rb vapor,” Phys. Rev. A 72, 065801 (2005).
[Crossref]

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V. A. Sautenkov, Y. V. Rostovtsev, and M. O. Scully, “Switching between photon-photon correlations and Raman anticorrelations in a coherently prepared Rb vapor,” Phys. Rev. A 72, 065801 (2005).
[Crossref]

M. O. Scully, K. Wódkiewicz, M. S. Zubairy, J. Bergou, N. Lu, and J. Myer ter Vehn, “Two-Photon Correlated-Spontaneous-Emission Laser: Quantum Noise Quenching and Squeezing,” Phys. Rev. Lett. 60, 1832–1835 (1988).
[Crossref] [PubMed]

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).

Seke, J.

Z. Ficek, J. Seke, A. V. Soldatov, G. Adam, and N. N. Bogoubov, “Multilevel coherence effects in a two-level atom driven by a trichromatic field,” Opt. Commun. 217, 299–309 (2003).
[Crossref]

Z. Ficek, J. Seke, A. V. Soldatov, and G. Adam, “Fluorescence spectrum of a two-level atom driven by a multiple modulated field,” Phys. Rev. A 64, 013813 (2001).
[Crossref]

Z. Ficek, J. Seke, A. V. Soldatov, and G. Adam, “Phase control of subharmonic resonances,” Opt. Commun. 182, 143–150 (2000).
[Crossref]

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P. B. Sellin, C. C. Yu, J. R. Bochinski, and T. W. Mossberg, “Intrinsically irreversible multiphoton laser gain mechanisms,” Phys. Rev. Lett. 78, 1432–1435 (1997).
[Crossref]

Shi, W. X.

X. M. Hu, Q. Xu, J. Y. Li, X. X. Li, W. X. Shi, and X. Zhang, “Bichromatic and trichromatic manipulation of spontaneous emission in a three-level system,” Opt. Commun. 260, 196–202 (2006).
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A. Sinatra, “Quantum correlations of two optical fields close to electromagnetically induced transparency,” Phys. Rev. Lett. 97, 253601 (2006).
[Crossref]

A. Sinatra, J. F. Roch, K. Vigneron, Ph. Grelu, J. Ph. Poizat, K. Wang, and P. Grangier, “Quantum-nondemolition measurements using cold trapped atoms: comparison between theory and experiment,” Phys. Rev. A 57, 2980–2995 (1998).
[Crossref]

Solano, E.

R. Guzmán, J. C. Retamal, E. Solano, and N. Zagury, “Field squeeze operators in optical cavities with atomic ensembles,” Phys. Rev. Lett. 96, 010502 (2006).
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A. S. Parkins, E. Solano, and J. I. Cirac, “Unconditional two-mode squeezing of separated atomic ensembles,” Phys. Rev. Lett. 96, 053602 (2006).
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Z. Ficek, J. Seke, A. V. Soldatov, G. Adam, and N. N. Bogoubov, “Multilevel coherence effects in a two-level atom driven by a trichromatic field,” Opt. Commun. 217, 299–309 (2003).
[Crossref]

Z. Ficek, J. Seke, A. V. Soldatov, and G. Adam, “Fluorescence spectrum of a two-level atom driven by a multiple modulated field,” Phys. Rev. A 64, 013813 (2001).
[Crossref]

Z. Ficek, J. Seke, A. V. Soldatov, and G. Adam, “Phase control of subharmonic resonances,” Opt. Commun. 182, 143–150 (2000).
[Crossref]

Stroud, C. R.

D. L. Aronstein, R. S. Bennink, R. W. Boyd, and C. R. Stroud, “Comment on Resonance-fluorescence and absorption spectra of a two-level atom driven by a strong bichromatic field,” Phys. Rev. A 65, 067401 (2002).
[Crossref]

S. Papademetriou, M. F. Van Leeuwen, and C. R. Stroud, “Autler-Townes effect for an atom in a 100% amplitude-modulated laser field. II. experimental results,” Phys. Rev. A 53, 997–1003 (1996).
[Crossref] [PubMed]

M. F. Van Leeuwen, S. Papademetriou, and C. R. Stroud, “Autler-Townes effect for an atom in a 100% amplitude-modulated laser field. I. a dressed-atom approach,” Phys. Rev. A 53, 990–996 (1996).
[Crossref] [PubMed]

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Yu. A. Zinin and N. V. Sushilov, “Absorption and dispersion spectra of a polychromatic field in a two-level medium driven by a strong polychromatic pumping field,” Phys. Rev. A 51, 3916–3921 (1995).
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B. Blind, P. R. Fontana, and P. Thomann, “Resonance fluorescence spectrum of intense amplitude modulated laser light,” J. Phys. B: At. Mol. Opt. Phys. 13, 2717–2727 (1980).
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C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sandberg, and Zimmermann, “On the rrieasureri-ient of a weak classical force coupled to a quantum-mechanical oscillator. I. Issues of principle,” Rev. Mod. Phys. 52, 341–392 (1980).
[Crossref]

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M. F. Van Leeuwen, S. Papademetriou, and C. R. Stroud, “Autler-Townes effect for an atom in a 100% amplitude-modulated laser field. I. a dressed-atom approach,” Phys. Rev. A 53, 990–996 (1996).
[Crossref] [PubMed]

S. Papademetriou, M. F. Van Leeuwen, and C. R. Stroud, “Autler-Townes effect for an atom in a 100% amplitude-modulated laser field. II. experimental results,” Phys. Rev. A 53, 997–1003 (1996).
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S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
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A. Sinatra, J. F. Roch, K. Vigneron, Ph. Grelu, J. Ph. Poizat, K. Wang, and P. Grangier, “Quantum-nondemolition measurements using cold trapped atoms: comparison between theory and experiment,” Phys. Rev. A 57, 2980–2995 (1998).
[Crossref]

Vitali, D.

S. Pielawa, G. Morigi, D. Vitali, and L. Davidovich, “Generation of Einstein-Podolsky-Rosen-entangled radiation through an atomic reservoir,” Phys. Rev. Lett. 98, 240401 (2007).
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E. G. Dalla Torre, J. Otterbach, E. Demler, V. Vuletic, and M. D. Lukin, “Dissipative preparation of spin squeezed atomic ensembles in a steady state,” Phys. Rev. Lett. 110, 120402 (2013).
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M. D. Reid, D. F. Walls, and B. J. Dalton, “Squeezing of quantum fluctuations via atomic coherence effects,” Phys. Rev. Lett. 55, 1288–1290 (1985).
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P. D. Drummond and D. F. Walls, “Quantum theory of optical bistability. II. atomic fluorescence in a high-Q cavity,” Phys. Rev. A 23, 2563–2579 (1981).
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D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1995).

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W. Lange, H. Walther, and G. S. Agarwal, “Decay of bichromatically driven atoms in a cavity,” Phys. Rev. A 50, R3593–R3596 (1994).
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G. S. Agarwal, W. Lange, and H. Walther, “Intense-field renormalization of cavity-induced spontaneous emission,” Phys. Rev. A 48, 4555–4568 (1993).
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J. Wang, Y. Zhu, K. J. Jiang, and M. S. Zhan, “Bichromatic electromagnetically induced transparency in cold rubidium atoms,” Phys. Rev. A 68, 063810 (2003).
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A. Sinatra, J. F. Roch, K. Vigneron, Ph. Grelu, J. Ph. Poizat, K. Wang, and P. Grangier, “Quantum-nondemolition measurements using cold trapped atoms: comparison between theory and experiment,” Phys. Rev. A 57, 2980–2995 (1998).
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H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, J. M. Petersen, J. I. Cirac, and E. S. Polzik, “Entanglement generated by dissipation and steady state entanglement of two macroscopic objects,” Phys. Rev. Lett. 107, 080503 (2011).
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A. D. Greentree, C. Wei, and N. B. Manson, “Polychromatic excitation of a two-level system,” Phys. Rev. A 59, 4083–4086 (1999).
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M. O. Scully, K. Wódkiewicz, M. S. Zubairy, J. Bergou, N. Lu, and J. Myer ter Vehn, “Two-Photon Correlated-Spontaneous-Emission Laser: Quantum Noise Quenching and Squeezing,” Phys. Rev. Lett. 60, 1832–1835 (1988).
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Y. Zhu, Q. Wu, A. Lezama, D. J. Gauthier, and T. W. Mossberg, “Resonance fluorescence of tvvo-level atoms under strong bichromatic excitation,” Phys. Rev. A 41, R6574–R6576 (1990).
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Y. Wu, M. G. Payne, E. W. Hagley, and L. Deng, “Preparation of multiparty entangled states using pairwise perfectly efficient single-probe photon four-wave mixing,” Phys. Rev. A 69, 063803 (2004).
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X. M. Hu, Q. Xu, J. Y. Li, X. X. Li, W. X. Shi, and X. Zhang, “Bichromatic and trichromatic manipulation of spontaneous emission in a three-level system,” Opt. Commun. 260, 196–202 (2006).
[Crossref]

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T. H. Yoon, M. S. Chung, and H. W. Lee, “Absorption spectra of two-level atoms interacting with a strong polychromatic pump field and an arbitrarily intense probe field,” Phys. Rev. A 60, 2547–2553 (1999).
[Crossref]

T. H. Yoon, S. A. Pulkin, J. R. Park, M. S. Chung, and H. W. Lee, “Theoretical analysis of resonances in the polarization spectrum of a two-level atom driven by a polychromatic field,” Phys. Rev. A 60, 605–613 (1999).
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Yu, C. C.

J. R. Bochinski, C. C. Yu, T. Loftus, and T. W. Mossberg, “Vacuum-mediated multiphoton transitions,” Phys. Rev. A 63, 051402(R) (2001).
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P. B. Sellin, C. C. Yu, J. R. Bochinski, and T. W. Mossberg, “Intrinsically irreversible multiphoton laser gain mechanisms,” Phys. Rev. Lett. 78, 1432–1435 (1997).
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C. C. Yu, J. R. Bochinski, T. M. V. Kordich, T. W. Mossberg, and Z. Ficek, “Driving the driven atom: spectral signatures,” Phys. Rev. A 56, R4381–R4384 (1997).
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Zagury, N.

R. Guzmán, J. C. Retamal, E. Solano, and N. Zagury, “Field squeeze operators in optical cavities with atomic ensembles,” Phys. Rev. Lett. 96, 010502 (2006).
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P. Barberis-Blostein and N. Zagury, “Field correlations in electromagnetically induced transparency,” Phys. Rev. A 70, 053827 (2004).
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Zhan, M. S.

J. Wang, Y. Zhu, K. J. Jiang, and M. S. Zhan, “Bichromatic electromagnetically induced transparency in cold rubidium atoms,” Phys. Rev. A 68, 063810 (2003).
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X. Zhang and X. M. Hu, “Entanglement between collective fields via atomic coherence effects,” Phys. Rev. A 81, 013811 (2010).
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X. M. Hu, Q. Xu, J. Y. Li, X. X. Li, W. X. Shi, and X. Zhang, “Bichromatic and trichromatic manipulation of spontaneous emission in a three-level system,” Opt. Commun. 260, 196–202 (2006).
[Crossref]

Zhong, W. X.

G. L. Cheng, X. M. Hu, W. X. Zhong, and Q. Li, “Two-channel interaction of squeeze-transformed modes with dressed atoms: entanglement enhancement in four-wave mixing in three-level systems,” Phys. Rev. A 78, 033811 (2008).
[Crossref]

Zhu, Y.

J. Wang, Y. Zhu, K. J. Jiang, and M. S. Zhan, “Bichromatic electromagnetically induced transparency in cold rubidium atoms,” Phys. Rev. A 68, 063810 (2003).
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G. S. Agarwal, Y. Zhu, D. J. Gauthier, and T. W. Mossberg, “Spectrum of radiation from two-level atoms under intense bichromatic excitation,” J. Opt. Soc. Am. B 8, 1163–1167 (1991).
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Y. Zhu, Q. Wu, A. Lezama, D. J. Gauthier, and T. W. Mossberg, “Resonance fluorescence of tvvo-level atoms under strong bichromatic excitation,” Phys. Rev. A 41, R6574–R6576 (1990).
[Crossref]

Zimmermann,

C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sandberg, and Zimmermann, “On the rrieasureri-ient of a weak classical force coupled to a quantum-mechanical oscillator. I. Issues of principle,” Rev. Mod. Phys. 52, 341–392 (1980).
[Crossref]

Zinin, Yu. A.

Yu. A. Zinin and N. V. Sushilov, “Absorption and dispersion spectra of a polychromatic field in a two-level medium driven by a strong polychromatic pumping field,” Phys. Rev. A 51, 3916–3921 (1995).
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L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
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Zou, J. H.

J. H. Zou, X. M. Hu, G. L. Cheng, X. Li, and D. Du, “Inhibition of two-photon absorption in a three-level system with a pair of bichromatic fields,” Phys. Rev. A 72, 055802 (2005).
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X. M. Hu, J. H. Zou, X. Li, D. Du, and G. L. Cheng, “Amplitude and phase control of trichromatic electromagnetically induced transparency,” J. Phys. B: At. Mol. Opt. Phys. 38, 683–692 (2005).
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X. M. Hu, G. L. Cheng, J. H. Zou, X. Li, and D. Du, “Double switching from normal to anomalous dispersion via trichromatic phase manipulation of electromagnetically induced transparency,” Phys. Rev. A 72, 023803 (2005).
[Crossref]

Zubairy, M. S.

M. O. Scully, K. Wódkiewicz, M. S. Zubairy, J. Bergou, N. Lu, and J. Myer ter Vehn, “Two-Photon Correlated-Spontaneous-Emission Laser: Quantum Noise Quenching and Squeezing,” Phys. Rev. Lett. 60, 1832–1835 (1988).
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M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).

J. Opt. Soc. Am. B (1)

J. Phys. A (1)

P. D. Drummond and C. W. Gardiner, “Generalised P-representations in quantum optics,” J. Phys. A 13, 2353–2368 (1980).
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J. Phys. B: At. Mol. Opt. Phys. (2)

B. Blind, P. R. Fontana, and P. Thomann, “Resonance fluorescence spectrum of intense amplitude modulated laser light,” J. Phys. B: At. Mol. Opt. Phys. 13, 2717–2727 (1980).
[Crossref]

X. M. Hu, J. H. Zou, X. Li, D. Du, and G. L. Cheng, “Amplitude and phase control of trichromatic electromagnetically induced transparency,” J. Phys. B: At. Mol. Opt. Phys. 38, 683–692 (2005).
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Opt. Commun. (3)

Z. Ficek, J. Seke, A. V. Soldatov, and G. Adam, “Phase control of subharmonic resonances,” Opt. Commun. 182, 143–150 (2000).
[Crossref]

Z. Ficek, J. Seke, A. V. Soldatov, G. Adam, and N. N. Bogoubov, “Multilevel coherence effects in a two-level atom driven by a trichromatic field,” Opt. Commun. 217, 299–309 (2003).
[Crossref]

X. M. Hu, Q. Xu, J. Y. Li, X. X. Li, W. X. Shi, and X. Zhang, “Bichromatic and trichromatic manipulation of spontaneous emission in a three-level system,” Opt. Commun. 260, 196–202 (2006).
[Crossref]

Phys. Rev. (1)

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[Crossref]

Phys. Rev. A (30)

A. Sinatra, J. F. Roch, K. Vigneron, Ph. Grelu, J. Ph. Poizat, K. Wang, and P. Grangier, “Quantum-nondemolition measurements using cold trapped atoms: comparison between theory and experiment,” Phys. Rev. A 57, 2980–2995 (1998).
[Crossref]

P. Barberis-Blostein and N. Zagury, “Field correlations in electromagnetically induced transparency,” Phys. Rev. A 70, 053827 (2004).
[Crossref]

Y. Wu, M. G. Payne, E. W. Hagley, and L. Deng, “Preparation of multiparty entangled states using pairwise perfectly efficient single-probe photon four-wave mixing,” Phys. Rev. A 69, 063803 (2004).
[Crossref]

X. Liang, X. M. Hu, and C. He, “Creating multimode squeezed states and Greenberger-Horne-Zeilinger entangled states using atomic coherent effects,” Phys. Rev. A 85, 032329 (2012).
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P. B. Li, “Generation of two-mode field squeezing through selective dynamics in cavity QED,” Phys. Rev. A 77, 015809 (2008).
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G. L. Cheng, X. M. Hu, W. X. Zhong, and Q. Li, “Two-channel interaction of squeeze-transformed modes with dressed atoms: entanglement enhancement in four-wave mixing in three-level systems,” Phys. Rev. A 78, 033811 (2008).
[Crossref]

X. M. Hu, “Entanglement generation by dissipation in or beyond dark resonances,” Phys. Rev. A 92, 022329 (2015).
[Crossref]

D. L. Aronstein, R. S. Bennink, R. W. Boyd, and C. R. Stroud, “Comment on Resonance-fluorescence and absorption spectra of a two-level atom driven by a strong bichromatic field,” Phys. Rev. A 65, 067401 (2002).
[Crossref]

Yu. A. Zinin and N. V. Sushilov, “Absorption and dispersion spectra of a polychromatic field in a two-level medium driven by a strong polychromatic pumping field,” Phys. Rev. A 51, 3916–3921 (1995).
[Crossref] [PubMed]

M. F. Van Leeuwen, S. Papademetriou, and C. R. Stroud, “Autler-Townes effect for an atom in a 100% amplitude-modulated laser field. I. a dressed-atom approach,” Phys. Rev. A 53, 990–996 (1996).
[Crossref] [PubMed]

S. Papademetriou, M. F. Van Leeuwen, and C. R. Stroud, “Autler-Townes effect for an atom in a 100% amplitude-modulated laser field. II. experimental results,” Phys. Rev. A 53, 997–1003 (1996).
[Crossref] [PubMed]

A. D. Greentree, C. Wei, and N. B. Manson, “Polychromatic excitation of a two-level system,” Phys. Rev. A 59, 4083–4086 (1999).
[Crossref]

T. H. Yoon, M. S. Chung, and H. W. Lee, “Absorption spectra of two-level atoms interacting with a strong polychromatic pump field and an arbitrarily intense probe field,” Phys. Rev. A 60, 2547–2553 (1999).
[Crossref]

T. H. Yoon, S. A. Pulkin, J. R. Park, M. S. Chung, and H. W. Lee, “Theoretical analysis of resonances in the polarization spectrum of a two-level atom driven by a polychromatic field,” Phys. Rev. A 60, 605–613 (1999).
[Crossref]

J. Wang, Y. Zhu, K. J. Jiang, and M. S. Zhan, “Bichromatic electromagnetically induced transparency in cold rubidium atoms,” Phys. Rev. A 68, 063810 (2003).
[Crossref]

X. M. Hu, G. L. Cheng, J. H. Zou, X. Li, and D. Du, “Double switching from normal to anomalous dispersion via trichromatic phase manipulation of electromagnetically induced transparency,” Phys. Rev. A 72, 023803 (2005).
[Crossref]

V. A. Sautenkov, Y. V. Rostovtsev, and M. O. Scully, “Switching between photon-photon correlations and Raman anticorrelations in a coherently prepared Rb vapor,” Phys. Rev. A 72, 065801 (2005).
[Crossref]

Y. Zhu, Q. Wu, A. Lezama, D. J. Gauthier, and T. W. Mossberg, “Resonance fluorescence of tvvo-level atoms under strong bichromatic excitation,” Phys. Rev. A 41, R6574–R6576 (1990).
[Crossref]

C. C. Yu, J. R. Bochinski, T. M. V. Kordich, T. W. Mossberg, and Z. Ficek, “Driving the driven atom: spectral signatures,” Phys. Rev. A 56, R4381–R4384 (1997).
[Crossref]

Z. Ficek and H. S. Freedhoff, “Resonance-Auorescence and absorption spectra of a two-level atom driven by a strong bichromatic field,” Phys. Rev. A 48, 3092–3104 (1993).
[Crossref] [PubMed]

Z. Ficek and T. Rudolph, “Quantum interference in a driven two-level atom,” Phys. Rev. A 60, R4245–R4248 (1999).
[Crossref]

Z. Ficek, J. Seke, A. V. Soldatov, and G. Adam, “Fluorescence spectrum of a two-level atom driven by a multiple modulated field,” Phys. Rev. A 64, 013813 (2001).
[Crossref]

G. Yu. Kryuchkyan, M. Jakob, and A. S. Sargsian, “Resonance fluorescence in a bichromatic field as a source of nonclassical light,” Phys. Rev. A 57, 2091–2095 (1998).
[Crossref]

M. Jakob and G. Yu. Kryuchkyan, “Squeezing in the resonance fluorescence of a bichromatically driven two-level atom,” Phys. Rev. A 58, 767–770 (1998).
[Crossref]

G. S. Agarwal, W. Lange, and H. Walther, “Intense-field renormalization of cavity-induced spontaneous emission,” Phys. Rev. A 48, 4555–4568 (1993).
[Crossref] [PubMed]

P. D. Drummond and D. F. Walls, “Quantum theory of optical bistability. II. atomic fluorescence in a high-Q cavity,” Phys. Rev. A 23, 2563–2579 (1981).
[Crossref]

J. R. Bochinski, C. C. Yu, T. Loftus, and T. W. Mossberg, “Vacuum-mediated multiphoton transitions,” Phys. Rev. A 63, 051402(R) (2001).
[Crossref]

J. H. Zou, X. M. Hu, G. L. Cheng, X. Li, and D. Du, “Inhibition of two-photon absorption in a three-level system with a pair of bichromatic fields,” Phys. Rev. A 72, 055802 (2005).
[Crossref]

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Figures (11)

Fig. 1
Fig. 1 (a) An ensemble of two-level atoms is placed at the intersection of two cavities, into which two external coherent fields are input respectively. (b) The two cavity fields (annihilation operators a1,2) of different frequencies are coupled to the common transition of the atoms.
Fig. 2
Fig. 2 The individual cavity fields a1,2 are both in resonant interactions with the dressed atoms. The two fields combine into the Bogoliubov mode b1 for Δ > 0 (N > N+) or b2 for Δ < 0 (N+ > N).
Fig. 3
Fig. 3 Approximately analytic variance 〈(δX)2〉 versus the scaled detuning Δ Ω . Within the shadow at Δ Ω = 0 , the conditions for approximately analytic solution are not well met.
Fig. 4
Fig. 4 The variance 〈(δX)2〉 versus the scaled detuning Δ Ω for different rates of cavity losses κ = 0.2γ (red dashed), κ = 0.1γ (black dotted), and κ = 0.01γ (blue solid). We have used the other parameter g N = 10 γ .
Fig. 5
Fig. 5 (a) Four optical cavities intersect at a common spot, where an atomic ensemble of three-level atoms is placed. (b) The interaction of the four cavity fields with the atoms in Λ configuration.
Fig. 6
Fig. 6 The dressed atomic transitions for N0 > N±. The left blue and red lines with arrows mean respectively the annihilation of the a1,2 modes and the creation of the a3,4 modes and constitute the collective b1 mode. The right red and blue lines with arrows show the annihilation of the a3,4 modes and the creation of the a1,2 modes and constitute the collective b2 mode.
Fig. 7
Fig. 7 (a) Interactions of the collective b1,2 modes with the dressed atoms via the transitions (i) from |0〉 to |±〉 (N0 > N±) (ii) from |±〉 to |0〉 (N± > N0). (b) The interactions between any two individual fields. All four fields are in a quadrilateral loop of two-photon interactions and in the X lines of two quantum-beat interactions. Any three fields are in a triangle of two two-photon interactions and one quantum-beat interaction.
Fig. 8
Fig. 8 Approximately analytic four-mode variance 〈(δX)2〉 versus the scaled detuning Δ Ω for a fixed phase Φ = 0. Four-mode squeezing occurs in almost entire region except at | Δ | Ω = 0 , 1 , 2 . Within two shadows at | Δ | Ω = 1 , the conditions for approximately analytic solution are not well met.
Fig. 9
Fig. 9 The numerical four-mode variance 〈(δX)2〉 versus the scaled detuning Δ Ω for different rates of cavity losses κ = 0.2γ (red dashed), κ = 0.1γ (black dotted), κ = 0.01γ (blue solid). The parameters are chosen as g N = 10 γ and Φ = 0.
Fig. 10
Fig. 10 The numerical four-mode variance 〈(δX)2〉 versus the scaled detuning Δ Ω for varying phases: Φ = 0 (blue dashed), Φ = π 2 (red dotted), Φ = 2 π 3 (black solid). The chosen parameters are g N = 10 γ and κ = 0.01γ.
Fig. 11
Fig. 11 The numerical four-mode variance 〈(δY)2〉 versus the scaled detuning Δ Ω for the same parameters as in Fig. 10.

Equations (48)

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ρ ˙ = i [ H , ρ ] + ρ ,
H = { Δ σ 11 + [ ( g 1 a 1 + g 2 a 2 ) σ 21 + H . c . ] } + j = 1 2 [ Δ c j a j a j + i ( ε j a j ε j * a j ) ] ,
ρ = γ 2 σ 12 ρ + j = 1 2 κ j 2 a j ρ ,
I in = I [ ( 1 + 𝒜 ) 2 + ( Δ ¯ c + 𝒟 ) 2 ] ,
𝒜 = 8 C 4 Δ ¯ 2 + 32 I + 1 , 𝒟 = 16 C Δ ¯ 4 Δ ¯ 2 + 32 I + 1 .
H = H a + H c + H I ,
H a = Δ σ 22 + 2 ( Ω σ 21 + Ω * σ 12 )
H c = Δ c ( δ a 1 δ a 1 δ a 2 δ a 2 )
| + = sin θ | 1 + cos θ | 2 | = cos θ | 1 sin θ | 2 ,
N + = N sin 4 θ cos 4 θ + sin 4 θ , N = N cos 4 θ cos 4 θ + sin 4 θ .
H I = g ( a 1 cos 2 θ a 2 sin 2 θ ) σ + + H . c . ,
b 1 = a 1 cosh r a 2 sinh r , b 2 = a 2 cosh r a 1 sinh r ,
H I = G ( b 1 σ + + σ + b 1 ) for Δ > 0 , H I = G ( b 2 σ + + σ + b 2 ) for Δ < 0 ,
ρ ˜ ˙ = 𝒦 a ρ ˜ + c ρ ˜ ,
𝒦 a ρ ˜ = A ( 2 b 1 ρ ˜ b 1 b 1 b 1 ρ ˜ ρ ˜ b 1 b 1 ) + B ( 2 b 1 ρ ˜ b 1 b 1 b 1 ρ ˜ ρ ˜ b 1 b 1 ) ,
β 1 a 1 ρ ˜ a 2 + β 2 ρ ˜ a 1 a 2 + β 3 a 1 a 2 ρ ˜ + ( 1 2 ) + H . c . ,
δ X = δ x a 1 δ x a 2 = e 4 ( δ x b 1 δ x b 2 ) , δ P = δ p a 1 + δ p a 2 = e r ( δ p b 1 + δ p b 2 ) .
( δ X ) 2 = e 2 r ( 1 + : ( δ x b 1 ) 2 : ) , ( δ P ) 2 = e 2 r ( 1 + : ( δ p b 1 ) 2 : ) ,
( δ X ) 2 = ( δ P ) 2 = e 2 r N + N N + for Δ > 0 , ( δ X ) 2 = ( δ P ) 2 = e 2 r N N + N for Δ < 0 .
α ˙ 1 = λ 11 α 1 + λ 12 α 2 * + F α 1 , α ˙ 2 = λ 21 α 1 * + λ 22 α 2 + F α 2 ,
2 D α 1 * α 1 = 2 g 2 Γ N + cos 4 θ , 2 D α 1 * α 2 = 2 g 2 Γ N sin 4 θ , 2 D α 1 α 2 = g 2 N 4 Γ sin 2 ( 2 θ ) , 2 D α 1 * α 2 * = 2 D α 1 α 2 .
H = j = 1 2 { Δ j σ j j + [ ( g j a j + g j + 2 a j + 2 ) σ 3 j + H . c . ] } + j = 1 4 [ Δ c j a j a j + i ( ε j a j ε j * a j ) ] ,
ρ = j = 1 2 γ j 2 σ j 3 ρ + j = 1 4 κ j 2 a j ρ ,
𝒜 = C Δ ¯ 2 Δ ¯ 4 + Δ ¯ 2 + Δ ¯ 2 I + I 2 , 𝒟 = C Δ ¯ ( Δ ¯ 2 + I ) Δ ¯ 4 + Δ ¯ 2 + Δ ¯ 2 I + I 2 .
H a = j = 1 2 [ ( 1 ) j + 1 Δ σ j j + ( Ω j + Ω j + 2 ) σ 3 j + ( Ω j * + Ω j + 2 * ) σ j 3 ]
H c = j = 1 2 Δ c ( δ a j δ a j δ a j + 2 δ a j + 2 )
| + = 1 + sin θ 2 | 1 + 1 sin θ 2 | 2 + cos θ 2 | 3 , | 0 = cos θ 2 | 1 + cos θ 2 | 2 + sin θ | 3 , | = 1 sin θ 2 | 1 + 1 + sin θ 2 | 2 cos θ 2 | 3 ,
N 0 = N cos 4 θ 1 + 3 sin 4 θ , N ± 1 2 ( N N 0 ) .
H I = 1 2 g [ cos 2 θ ( a 1 a 2 ) + sin θ ( 1 + sin θ ) a 3 + sin θ ( 1 sin θ ) a 4 ] σ + 0 + 1 2 g [ cos 4 θ ( a 3 a 4 ) + sin θ ( 1 sin θ ) a 1 + sin θ ( 1 + sin θ ) a 2 ] σ 0 + H . c . ,
b 1 = p ( a 1 a 2 ) u a 3 v a 4 , b 2 = p ( a 3 a 4 ) + v a 1 + u a 2 , b 3 = w a 3 + z a 4 q ( a 1 a 2 ) , b 4 = z a 1 + w a 2 + q ( a 3 a 4 ) ,
H = G ( b 1 σ + 0 + b 2 σ 0 ) + H . c . ,
𝒦 a ρ ˜ = A ( b 1 ρ ˜ b 1 b 1 b 1 ρ ˜ + b 2 ρ ˜ b 2 b 2 b 2 ρ ˜ ) + B ( b 1 ρ ˜ b 1 b 1 b 1 ρ ˜ + b 2 ρ ˜ b 2 b 2 b 2 ρ ˜ ) + D ( b 1 ρ ˜ b 2 + b 2 ρ ˜ b 1 ) D 1 ( ρ ˜ b 1 b 2 + ρ ˜ b 2 b 1 ) D 2 ( b 1 b 2 ρ ˜ + b 2 b 1 ρ ˜ ) + H . c . ,
β 1 a k ρ ˜ a l + β 2 ρ ˜ a k a l + β 3 a k a l ρ ˜ + ( k l ) + H . c . ,
β 1 a k ρ ˜ a l + β 2 ρ ˜ a l a k + β 3 a l a k ρ ˜ + ( k l ) + H . c . ,
a 1 𝕋 a 3 𝕋 a 2 𝕋 a 4 𝕋 a 1 .
a 1 𝔹 a 2 , a 3 𝔹 a 4 .
a 1 𝕋 a 3 𝕋 a 2 𝔹 a 1 , a 1 𝕋 a 4 𝕋 a 2 𝔹 a 1 , a 3 𝕋 a 1 𝕋 a 4 𝔹 a 3 , a 3 𝕋 a 2 𝕋 a 4 𝔹 a 3 .
b 1 = cosh r a 1 a 2 2 ξ sinh r u a 3 + v a 4 u 2 + v 2 ,
r = arctanh | sin θ 1 + sin 2 θ | cos 2 θ for | Δ | Ω ¯ < 1 , r = arctanh cos 2 θ | sin θ 1 + sin 2 θ | for | Δ | Ω ¯ > 1 ,
X = x a 1 x a 2 2 ξ u x a 3 + v x a 4 u 2 + v 2
( δ X 2 ) = e 2 r ( 1 + : ( δ x b 1 ) 2 : + sin 2 θ 1 + sin 2 θ : ( δ x b 2 ) 2 : ) ,
( δ X ) 2 = e 2 r ( 1 + 1 + 2 sin 2 θ 1 + sin 2 θ N + N 0 N + ) .
( δ X ) 2 = e 2 r ( 1 + 1 + 2 sin 2 θ 1 + sin 2 θ N 0 N + N 0 ) .
P = p a 1 p a 2 2 + ξ u p a 3 + v p a 4 u 2 + v 2 , X = x a 3 x a 4 2 + ξ v x a 1 + u x a 2 u 2 + v 2 , P = p a 3 p a 4 2 ξ v p a 1 + u p a 2 u 2 + v 2 ,
α ˙ 1 = λ 11 α 1 + λ 12 α 2 + λ 13 α 3 * + λ 14 α 4 * + F α 1 , α ˙ 2 = λ 21 α 1 + λ 22 α 2 + λ 23 α 3 * + λ 24 α 4 * + F α 2 , α ˙ 3 = λ 31 α 1 * + λ 32 α 2 * + λ 33 α 3 + λ 34 α 4 + F α 3 , α ˙ 4 = λ 41 α 1 * + λ 42 α 2 * + λ 43 α 3 + λ 44 α 4 + F α 4 ,
λ 11 = λ 44 = κ / 2 ( A B ) β 1 , λ 22 = λ 33 = κ / 2 ( A B ) β 2 , λ 23 = λ 32 = ( D 1 D 2 ) β 2 , λ 14 = λ 41 = ( D 1 D 2 ) β 1 , λ 13 = λ 42 = ( A B ) cos θ sin ( 2 θ ) + ( D 1 D 2 ) cos 2 θ , λ 31 = λ 24 = ( A B ) cos θ sin ( 2 θ ) + ( D 1 D 2 ) cos 2 θ , λ 12 = λ 43 = ( A B ) cos 2 θ + ( D 1 D 2 ) cos θ sin ( 2 θ ) , λ 21 = λ 34 = ( A B ) cos 2 θ ( D 1 D 2 ) cos θ sin ( 2 θ ) ,
D α 1 * α 1 = A η 1 2 + B cos 4 θ D η 1 cos 2 θ , D α 3 * α 3 = A η 2 2 + B cos 4 θ + D η 2 cos 2 θ , D α 1 α 4 = ( A + B ) η 1 cos 2 θ D 1 cos 4 θ D 2 η 1 2 , D α 1 α 3 = ( A + B D 2 ) cos 2 θ sin 2 θ + D 1 cos 4 θ , D α 2 α 3 = ( A + B ) η 2 cos 2 θ D 1 cos 4 θ D 2 η 2 2 , D α 1 α 2 * = ( A sin 2 θ B cos 2 θ ) cos 2 θ D cos 2 sin 2 θ , D α 2 * α 2 = D α 3 * α 3 , D α 4 * α 4 = D α 1 * α 1 , D α 2 α 4 = D α 1 α 3 , D α 3 α 4 * = D α 1 α 2 * ,
Y = 1 2 [ x a 1 x a 2 ξ ( x a 3 + x a 4 ) ] ,

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